Problem: Write the equation for a parabola with a focus at $(6,7)$ and a directrix at $x=1$. $x=$
The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(6,7)$, is equal to the distance between $(x,y)$ and the directrix, $x=1$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(6,7)$ is $\sqrt{(x-6)^2+(y-7)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $x=1$ is $\sqrt{(x-1)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(x-1)^2} &= \sqrt{(x-6)^2+(y-7)^2} \\\\ (x-1)^2 &= (x-6)^2+(y-7)^2 \\\\ {x^2}-2x{+1} &= {x^2}{-12x}+36+(y-7)^2\\\\ -2x{+12x}&=(y-7)^2+36{-1} \\\\ 10x&=(y-7)^2+35\\\\ x&=\dfrac{(y-7)^2}{10}+\dfrac{7}{2}\end{aligned}$ The answer The equation of our parabola is $x=\dfrac{(y-7)^2}{10}+\dfrac{7}{2}$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${10}$ ${11}$ ${12}$ ${13}$ ${14}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ $y$ $x$ ${(x,y)}$